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Experimental and numerical study on the behavior of concrete subjected to biaxial tension and shear
Experimental and Numerical Study on the Behavior of Concrete Subjected to Biaxial Tension and Shear M.B. Nooru-Mohamed, Erik Schlangen, and Jan G.M. van Mier Stevin Laboratory, Faculty of Civil Engineering, Delft University of Technology, Delft, The Netherlands
In this article an experimental and numerical study on the behavior of concrete subjected to biaxial loading is outlined. For this purpose the unique biaxial machine available at the Stevin Laboratory was used. Two load-paths were pursued, namely axial tension at constant shear and proportional tension~shear. The recently developed lattice model was used to simulate the two load-paths. The remarkable feature of the lattice model is its ability to simulate curved overlapping cracks which resemble the experimental findings. ADVANCEDCEMENT BASED MATERIALS 1993, 1, 22-37 KEY WORDS: Biaxial loading, Concrete, Fracture, Lattice
model, Load-paths, Shear, Tension, Testing
~
oncrete fracture studies have been pursued in r e c e n t years with great enthusiasm. As a result, the theoretical and experimental knowledge related to mode I fracture of concrete attained its zenith. It was often argued that the real-life problems are seldom confined to mode I fracture alone, and also most structural situations are a combination of tearing (mode I) and shearing (mode II) or twisting (mode III) modes, known as mixed-mode in the classical fracture mechanics terminology. As a natural consequence researchers have attempted to extend the concept of mode I to mixed-mode fracture of concrete. However, to what extent failure is governed by mixed-mode fracture is not straightforward. In this article the results of an extensive study on the behavior of plain concrete subjected to combined tensile (mode I) and in-plane shear loading (mode II) are presented. The intention of most fracture mechanics research of concrete has been the determination of material fracture properties for numerical models in structural analysis. The macroscopic p h e n o m e n o n observed in an ex-
m
Address correspondence to: Jan G.M. van Mier, Stevin Laboratory, Faculty of Civil Engineering, Delft University of Technology, P.O. Box 5048, 2600 6A Delft, The Netherlands. © Elsevier Science Publishing Co., Inc. ISSN 1065-7355/93/56.00
periment is to a large extent due to the processes occurring at a lower level [1]. The presence of aggregative constituents is the main cause of heterogeneity in concrete. Based on the ratio between the maximum aggregate size and the characteristic dimension of a specimen, the structure (specimen) can be treated either as homogeneous or as heterogeneous. At the plastic and hardened stage, nonstructural intrinsic cracks form due to the nature of the various constituents of concrete. Therefore, cracks exist at the aggregate mortar interface even before any mechanical load is applied [2]. Two regions, prepeak and postpeak, can be identiffed on a macroscopic global load-deformation (P-8) curve of a uniaxial tensile bar. In the prepeak region micro cracks evolve, but in comparison to the heterogeneity of the material, these cracks are relatively small and w h e n a representative volume is considered, the continuum theory is valid in this region. However, in the postpeak region, conveniently referred to as softening region, discrete cracks form. Here, the size of the discrete crack is of the same order of magnitude as that of the specimen, and the continuum theory is violated. The final global response of the experiment is a combination of the response of the material, the geometry of the specimen, and the boundary conditions under which the test is carried out. It is important to distinguish between the material and structural properties in a (fracture) test. However, these differences cannot be identified in a straightforward manner. It is doubtful whether one can ever obtain the real material fracture properties without the various influences, such as the boundary conditions, the specimen shape/size, etc [3]. The capacity of a partially opened crack to transfer shear is of practical interest. The curved crack which forms in reinforced concrete beams or columns [4] and cracks growing in the upstream face of a gravity dam [5] are two such examples. When the crack openings are small, the transfer of shear is not only associated with aggregate interlock mechanism but is also related Received February 9, 1993 Accepted May 25, 1993
Advn Cem Bas Mat 1993;I :22-37
Biaxial Tension and Shear on Concrete
to fracturing of the aggregate mortar matrix. Attempts were made in refs 6 and 7 to link the aggregate interlock theories for cracks loaded in simultaneous tension and shear. The existing aggregate interlock models fall short of capturing the actual phenomenon when the crack openings are small. The second effect, fracturing of the aggregate mortar matrix, is normally neglected. The purpose of the present fracture studies is to broaden the knowledge to the situation where tension and shear act simultaneously. The unique biaxial machine [8-10] built at the Stevin Laboratory of Delft University of Technology was used. The machine consists of two independent frames which are capable of moving along the vertical and horizontal directions. Axial load and lateral shear are applied to the specimen through this configuration of independent loading frames. Double-edge-notched (DEN) specimens of three different sizes are used. Experimental research and numerical tools must play a complementary role to arrive at a better understanding of the behavior of concrete. On the one hand numerical tools can be utilized to interpret the experi-
mental observations, and on the other hand experimental results can be used to tune a numerical model. Because the fracture process is a discrete phenomenon, the continuum-based approaches fall short of adequately describing this behavior. Therefore, a more convincing approach might be the use of micromechanics-based models to study fracture processes. In this article the experimental observations of two load-paths are discussed, namely axial tension at constant shear (load-path 1) and proportional tension/ shear (load-path 2). Formation of curved overlapping cracks is the main feature of load-path 1. The recently developed lattice model [11,12] was used to simulate load-paths 1 and 2. The remarkable feature of the lattice model is its ability to simulate curved overlapping cracks which resemble exactly the experimental findings. Further comparison is made between results of the numerical simulation and the experimental loaddisplacement curves. It seems that the mode I criterion adopted in the lattice model is sufficient for simulating the biaxial tension/shear behavior observed in the experiments.
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24 Nooru-Mohamed et al.
Advn Cem Bas Mat 1993;1:22-37
Testing Technique In this section a summary of the experimental technique is given. For a more extensive description of the biaxial testing machine and the test technique, refer to ref 10.
Materials and Specimens In this study, DEN square concrete elements of three different sizes were used; namely 200 x 200, 100 x 100, and 50 x 50 mm, with equal thickness of 50 m m (each having a constant notch to depth ratio a/d = 0.125). The maximum aggregate size was 2 mm. Details of the concrete mix are given in the Appendix.
Loading Frame To study the behavior of concrete under biaxial tension/shear loading, a unique machine was developed at the Stevin Laboratory of Delft University of Technology. The biaxial machine consists of two independent, stiff, square loading frames. The exploded view of the machine is shown in Figure 1. The outer frame A is a coupled frame and it is capable of moving along the horizontal direction. The inner frame B can move along the vertical direction. Moreover, frames A and B can move independently of one another. The outer frames and the inner frame are fixed to the overall frame via eight vertical and four horizontal plate springs. The assumption is that the plate springs pre(a)
plote springs
(b)
FIGURE 2. Photograph of the biaxial test set-up. vent the frames from rotating. They have a limited bending capacity. On each side of the plate spring two strain gauges are mounted and connected to the data acquisition system. Any eccentricity which may occur during a test in the frame assembly could be traced back from the plate spring readings. Two hydraulic actuators (A,B) and two load cells (A,B) are situated outside the loading frames so as not to hinder the overall stiffness of the loading frame. The frame assembly is shown in Figure 2. In Figure 3a and b, the sectional view of the loading middle frome
bending of plote springs
connecting plote for coupled fromes
FIGURE 3. Sectional view of the biaxial loading frames in two stages: (left) before and (right) after the specimen is cracked.
Advn Cem Bas Mat 1993;1:22-37
Biaxial Tension and Shear on Concrete
frame
either the average vertical deformation ~ or the average global shear deformation 8s was used as the feedback signal in the closed loop servo control system.
B
Measurements and Data Acquisition
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us
25
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FIGURE 4. Boundary condition for a DEN specimen, subjected to combined tension and shear. frame is shown in two stages. Here a DEN specimen is glued to the frames. As can be seen from a and b, the top of the specimen is glued to the inner frame and the bottom of the specimen to the coupled outer frame. The inner frame can slide in between the coupled frame, as shown in a. The specimen is loaded by moving the middle frame upward. A tensile stress develops in the concrete specimen because the coupled outer frames are fixed in the vertical direction. A tensile crack always nucleates in the notched region, which is located at half height of the DEN specimen (Figure 3b), so that crack evolution can be followed at a known location. This enables deformation controlled testing. Note the b e n d i n g of the horizontal plate springs in Figure 3b after crack formation in the DEN specimen. The DEN specimens are fixed to the frames via loading platens. For a uniaxial tensile test only two loading platens are required. However, for a biaxial test two additional loading platens are needed to apply a lateral in-plane shear load. The assumed boundary condition for a DEN specimen is outlined schematically in Figure 4. The boundary conditions allow for parallel end-plate movement in the vertical and horizontal directions during the test.
Test Control The mixed-mode machine has two independent frames connected to two independent regulation circuits, and it is therefore possible to control the vertical and horizontal deformations separately in a test. Moreover, tests can also be performed with deformation control in the vertical direction and load control in the horizontal direction and vice versa. The average signal of the control LVDTs was amplified via a standard amplifier (SCHENCK MV 318) to the servo controller unit. A function generator was connected to the servo controller. Depending on the type of test, the signals of
The signals related to the movement of the two loading frames A (the outer frame, capable of moving in the horizontal direction) and B (the inner frame, which can move in the vertical direction) were connected to two independent regulation circuits 1 and 2 of the SCHENCK regulation system. The machine contains two actuators and two load-cells (see Figure 1). The capacity of the loading axes is restricted to 50 kN in the vertical and horizontal directions because of the limited strength of the plate springs. For the 200- and 100-ram specimens, four LVDTs (Sangamo A6G1, gauge length = 65 ram), and for the 50-ram specimen two LVDTs (gauge length = 35 ram) were used. The LVDTs were calibrated using a device containing a Mitutoyo micro-screw with 0-50-ram movement and a 5-~m accuracy. The LVDT has a linear stroke of --1 m m with a nonlinearity of 0.3%. For the lateral d e f o r m a t i o n m e a s u r e m e n t s two LVDTs (Sangamo, A6G1) were used in the horizontal direction. At the front face of the 200-ram specimen, one LVDT was mounted at the center of the bottom half, as shown in Figure 4, the second LVDT was fixed to the diagonally opposite position at the rear face of the specimen. The global shear displacement ~s was defined as the average displacement between the specimen halves measured by the two LVDTs. In addition to the LVDTs, a number of locally built extensometers were used to measure local deformations. The extensometer has a range of 0 to 2 ram, a resolution of 1 p,m, a nonlinearity of 0.5%, and an accuracy of 5 p,m. The same calibration device mentioned above was used to calibrate the extensometers. The number of extensometers used varied depending on the type of test and the size of the specimen. A 32 channel programmable data logger (designed by the Measurement and Instrumentation Group of the Stevin Laboratory) was used to retrieve the data from the extensometers and strain gauges. The total scan t~'ne for 32 channels is 976 ms. A micro computer was required to program the data logger. Dedicated software was developed that allowed fully automatic testing. The personal computer controls the data logger, the SCHENCK regulation system, the A/D convertor, and the function generator. During a test scanning was done at fixed time intervals. The load was applied continuously and no stops were made for scanning of the measuring devices. The data logger was programmed to scan every 10 seconds from the beginning of a test, with the capability of changing the scan time (usually 30 to 60 seconds). A
26
Nooru-Mohamed et al.
Advn Cem Bas Mat 1993;1:22-37
typical scan consists of signals from two loads (P, P~) [mV], two deformations (6,5s) [mV], six to 20 extensometers [I~V], and 12 strain gauges [txV] from the vertical and horizontal plate springs.
Results and Discussions Load-Path Description In this article the direction of the lateral shear load is shown in Figure 4. When referring to the crack opening, it will be implied that no correction was made for the linear elastic deformation within the measuring length of the LVDT or extensometer. In Figure 5, the two load-paths investigated are shown schematically as P-P~, P-6, and 6-time diagrams. They are the following: LOAD-PATH 1
Axial Tension at a Constant Shear Force (Figure 5a). A compressive shear load Ps was applied to the specimen in displacement control until Ps = - 5 kN, - 1 0 kN,
and to the maximum shear load the specimen could sustain for load-path la, lb, and lc, respectively. LOAD-PATH 2
Proportional Loading (Figure 5b). An axial tensile and a lateral compressive shear load were applied to the specimen such that the ratio of axial (5) to lateral deformation (Ss) remained constant throughout the test. The 6/Ss-ratio for load-path 2a, 2b, and 2c was 1.0, 2.0, and 3.0, respectively.
Experimental Results and Numerical Simulations INFLUENCE OF LATERAL CONFINEMENT (LOAD-PATH 1)
Experiments. In the load-path 1 experiments, first a lateral compressive shear was applied in displacement control. Subsequently, the lateral test control was changed to load control and an axial tensile load was applied in displacement control. In other words, a constant lateral confinement was maintained while the ax-
Ps = -5,-lO, P~,mo
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FIGURE5. Schematic representation of load-paths I and 2. In (a) load-path 1, first shear is applied (P, = -5, - 10, and maximum shear in load-path la, b, and c, respectively), followed by axial tension at constant P,; In (b) load-path 2 proportional 8/8~ is applied.
Advn Cem Bas Mat 1993;1:22-37
Biaxial Tension and Shearon Concrete 27
(kN) 0 []
P , = - 5 kN ILp a ) ( 4 8 - 0 3 ) P , = - I O kN Lp I b ) ( 4 6 - 0 5 ) P , = - I O kN fLP 11b)(47-01 ) Ps= max Lp c ) ( 4 7 - 0 6 )
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the 8i-~s plot (Figure 7a), the largest increase in 8i first occurred at gauges 1 and 2 and subsequently at gauge 5. In Figure 7b, the locations of the measuring points are identified. This confirms the crack initiation at the left and right notches. The crack opening was faster at the left notch than at the right notch. Evidently, this crack initiation occurred during subsequent axial pulling at a constant lateral compressive shear (Ps) equal to - 1 0 kN. The two overlapping curved cracks EF and GH with a compressive strut (i.e., the enclosed area between EF and GH) in the middle were formed at the end of the test, as shown in Figure 8b. Specimen 48-03 subjected to load-path la behaved in a similar fashion. However, the overlapping cracks AB and CD (Figure 8a) were more fiat, which resulted in a smaller strut size. For the specimen 47-06 which was first loaded up to maximum shear, the Ps-Bs diagram became nonlinear at P~ = - 12.9 kN and Bs = - 30 I~m (Figure 6a). This nonlinearity is due to the crack initiation at the front in
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FICURE 6 . (a) Ps-$s and (b) P-8 response for 200-mm specimen corresponding to load-path 1 experiments.
0 -5
ial tensile force was activated. In Figure 6a and b, the P,-B, and the P-B diagrams for the 200-mm specimens corresponding to load-path la, b, and c are shown. The shear load for load-path la, b, and c was - 5 , - 1 0 , and -27.5 kN (maximum shear), respectively. For specimen 46-05 (Figure 6a), the Ps-8~ plot is linear until P~ = - 10 kN. At P, = - 10 kN, the shear deformation, 8s, and the highest local deformation, Bi (Figure 7), corresponding to the measuring point 1 were - 2 2 I~m and 4 I~m, respectively. At point B on the Ps-~s diagram where ~ = - 2 2 I~m for load-path lb (Figure 6a), the tensile load is activated. As can be seen
I
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-20
I
-30
I
-40
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(b) F I C U R E 7. (a) Local deformation versus lateral shear (8i-8~) and the (b) location of measuring points for specimen 46-05 (load-path lb).
2 8 Nooru-Mohamed et
al.
Advn Cem Bas Mat 1993;1:22-37
front face
(a) rear face
load-path la(48-03) front face
(b) rear face
load-path lb(46-05) front face
L
(c)
load-path lc(47-06)
FIGURE 8. Crack patterns for the lateral confinement experiments (load-path 1) of Figure 6. left notch (measuring point 1), as can be seen in the 8i-8s plot in Figure 9a. Further shearing led to a maxim u m shear load of - 27.5 kN, at which time the shear deformation was - 9 2 . 4 p,m. As can be seen in Figure 9, the largest increase in 8 i occurred first at gauge 1, subsequently at gauges 11, 6 (811 = 86), and 7. Gauges 1 and 11 were located near the left notch at the front and rear faces of the specimen. Evidently, the crack propagated on the front face of the specimen from the left notch to the location of gauge 7. This crack extension is identified as IJ on the crack pattern shown in Figures 8c and 9c. However, on the rear face the fracture propagation clearly lagged behind during the application of the lateral shear load. The crack on the rear face reached gauge 17 (located opposite to gauge 7) only after activating the axial tension. The crack prop-
agation during the lateral shear caused an increase of (note that the LVDTs were mounted near the notches). This can clearly be seen from Figure 6b: point A on the P-~ diagram for load-path lc corresponds to the axial deformation during the application of the lateral compressive shear load. (In the shear regime, load-control was maintained in the axial direction and P = 0 = constant.) At A the axial tensile loading was activated, maintaining a constant lateral shear load of - 2 7 . 5 kN (slightly below the maximum shear capacity of the specimen). As can be seen in Figure 6b, the axial load P became compressive almost immediately after it was activated. The crack KL initiated at the right notch. Thus, during crack propagation a small axial compressive load had to be maintained to obtain a stable test result. The highest axial tensile load (15 kN) was measured in the test with the low shear load of - 5 kN. A 30% decrease in P occurred when Ps was increased from - 5 kN to - 1 0 kN. Clearly the presence of a lateral compressive shear load (or a lateral confinement) significantly affected the axial tensile capacity of the specimen. Figure 8 shows the crack patterns for the loadpath 1 experiments. A relation seems to exist between the lateral shear load (or lateral confinement) and the area of the compressive struts. The specimen with the maximum shear load (Figure 8c) shows the largest strut area, whereas the specimen with the lowest shear (Figure 8a) gives the smallest strut size. In all the specimens the crack initiated near the notch at an angle of approximately 45 °. Owing to the application of the lateral compressive shear and the axial tension, the stress state changes at the crack tip and the principal stresses rotate. Due to this principal stress rotation, the crack propagates along a curvilinear path. To check the repeatability of the test results a small number of tests were duplicated in a random manner. As can be seen in Figure 6b, two tests of Ps = - 10 kN were repeated. There exist some deviations in the prepeak behavior of the two specimens that can likely be attributed to the difference in the age of loading (48 and 157 days, respectively). However, self similar crack patterns were observed for both specimens. Numerical simulation of fracture processes. Numerical tools can be used to gain some insight into the fracture behavior of concrete structures. Experimental research and numerical tools must play a complementary role to arrive at a better understanding of the behavior of complex disordered materials like concrete. On the one hand numerical tools can be utilized to interpret the experimental observations, and on the other hand experimental results can be used to tune a numerical model. The fracture process in concrete is a discrete phenomenon. The continuum-based approaches fall short
Advn Cern Bas Mat 1993;1:22-37
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Biaxial Tension and Shear on Concrete
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front; front front; front;
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of capturing or adequately describing this behavior. The formation of a macro crack depends to a large extent on processes occurring at a lower (micro-mesa) level of the material. Therefore, a more convincing approach might be the use of micro-mechanics-based models to study fracture processes. The recently proposed lattice model (for details, refer to refs 11 and 12) is used to simulate the two load-paths (1 and 2) discussed in this article. In the following paragraphs, a comparison is made between the numerical simulation and the experimental results. In the lattice model the material is schematized as a lattice of brittle breaking beam elements. A triangular lattice is chosen for convenience. Three options for intraducing disorder in the lattice model have been proposed thus far. In the first approach, strength and/or stiffness values drawn from a statistical distribution are assigned to the individual beam elements in a regular lattice. Second, the lattice can be overlaid on top of a g e n e r a t e d grain s t r u c t u r e of concrete. Different strength and stiffness values are assigned to beam el-
ements overlapping with the aggregate particles, the matrix material, and the bond zone. Third, a random lattice with a variable beam length can be used with a constant beam strength. A simple fracture law is implemented in the lattice model. Fracture is simulated by removing in each loadstep the element whose stress has exceeded the assigned tensile strength. A linear finite element analysis is only required to evaluate the stress, and at each load step the above procedure is repeated. It seems that softening can be omitted altogether in this approach. However, a parameter study is required before simulating various load-paths. Accordingly, the relevant parameters of the lattice model were tuned based on the uniaxial tensile experiments [12]. Note that only a small number of single valued parameters suffice for describing the local material properties. SIMULATION OF LOAD-PATH 1 Tension at a Constant Shear. In Figure 10a and b, the
simulated Ps-Ss and P-8 plots for load-path 1 are shown. The simulations were stopped at 8s ~ 60 ~m. According to the load-path applied in the experiments, the shear loads were kept constant at - 5 kN and - 10 kN for path la and lb, respectively. The simulated crack patterns are shown in Figure 11. For the loadpath 1 simulations, one specimen size (viz. 200 mm) was used. The lattice model with a variable beam length was adopted in these analyses. In line with the observations, the formation of the
30 Nooru-Mohamed et al.
Advn Cem Bas Mat 1993;1:22-37
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FIGURE 10. (a) Ps-~s and (b) P-8 response for the 200-mm specimen corresponding to the simulation of load-path 1. curved overlapping cracks was found in the simulations. The remarkable feature of the lattice model is that it can capture the curved cracks. Contrary to this, continuum-based approaches such as the smeared crack model fail to produce the curved cracks; also, the crack often forms along the boundaries of the element mesh [13,14]. A (qualitative) comparison between the simulated and the experimental crack patterns shows a close resemblance (compare Figures 8 and 11). An in-
crease in the lateral confinement produced an increase in the confined area between the overlapping cracks. The same observation was made in the experiment too (see Figure 8). For load-path la, the simulated Ps-~s diagram exactly matches the experimental result. Also, the simulated peak shear load (load-path lc) is identical to that of the experimental value. However, for load-path lb and c with Ps = - 10 kN and maximum shear, respectively, the simulated Ps-~s diagrams show a higher stiffness as compared to the experimental results (see Figures 10a, and 6a). This is caused by the additional plane stress elements attached to the halves of the mesh edges A and B (see Figure 11a and b). These additional plane stress elements with an equivalent concrete stiffness were provided for the simulations with Ps = - 10 kN and the maximum shear only. The addition of stiff elements led to a higher shear load, and therefore a visible higher stiffness of the initial part of the P~-~ diagram. The purpose of adding these stiff elements was to prevent premature cracking along the specimen loading interface. However, this goal was only partially achieved because, at the peak shear load, cracks still formed along the boundary. To avoid the formation of boundary cracks at the peak shear load, simulations were repeated for load-path lc with an additional column of plane stress elements with a stiffness equivalent to that of steel. The influence of the loading platen stiffness on crack initiation can clearly be seen in Figure 11c and d. Both simulated crack patterns correspond to the situation at maximum compressive shear load. For the mesh with lower stiffness (Figure 11c), in addition to the familiar overlapping cracks, boundary cracks as well as strut-area cracks formed. However, for the case with a higher loading platen stiffness (in fact resembling the well-known shear box experiments), the additional cracks near the loading platen did not appear (see Figure 11d). This indicates that a shear box type loading reduces stress concentrations at the two corners (top right/bottom left) of the specimen. In that case, the formation of glue layer cracks is unlikely w h e n the maximum shear force is applied. In the experiments the occurrence of additional boundary cracks was not seen for large specimen sizes, namely 200 ram. However, the above phenomenon was frequent in the case of specimens with a thickness of 25 m m instead of 50 mm. It seems that the out-ofplane stiffness is not adequate for a stable test. Moreover, the boundary cracks were also prevalent w h e n small specimens (viz. 100 and 50 m m with normal thickness of 50 ram) were used. The individual plate spring reading would show some deviations if there were glue layer cracks. Hence, the vertical load transfer would not be uniform.
Biaxial Tension and Shear on Concrete 31
Advn Cem Bas Mat 1993;1:22-37
A
B
(b)
(a)
(c)
V I I I I I I I I I I J I I I I I I I I I I ) I [ I I J I I I I I I I I I ] I
(d)
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j-r-~, ~I I I I l l l l t l i l ~l l l ] l l Il I [ l l l l l l l l l [ l l [ l l rl - c a a c a P ~ , ,~
FIGURE 11. Simulated crack patterns with random lattice: (a) Ps = - 5 kN, (b) Ps = - 1 0 kN with additional column of plane stress elements at edges A and B, (c) Ps = maximum shear with mesh as in (b), and (d) Ps = maximum shear with two columns of plane stress elements with high stiffness.
Proportional Loading (Load-Path 2) INFLUENCE OF THE 8/8 s RATIO FOR THE 200 M M SPECIMEN
Experiment. In load-path 2, the axial tensile a n d the lateral shear load w e r e applied such that the 8/8s ratios were constant a n d equal to 1.0, 2.0, a n d 3.0. The spec-
imen size was 200 m m . Figure 12a a n d b s h o w s the P-8 a n d Ps-Ss plots for the 200-mm specimen. The influence of the 8/8s ratio is seen in these plots. For 8/8s = 1.0, the p o s t p e a k P-8 curve w e n t into c o m p r e s s i o n as c o m p a r e d to the o t h e r two ratios. Moreover, the 8/8~ = 1.0 p r o d u c e d the highest lateral shear. The influence of the lateral shear is m o r e d o m i n a n t w h e n the axial-
32 Nooru-Mohamed et al.
Advn Cem Bas Mat 1993;1:22-37
P (kN) o [] z~
ps (kn)
-40.0
20.0
6/8,=I.0 (Lp 20)(47-02) 5Z6~=I.0 (Lp 2a)(46-02~ 6~6,--210 {,Lp 2b){48-01)
:::)6/6s= I .0(Lp20)(47-02) u 6Z8.= 1.0(Lp2a)(46-02.) ,, 6 Z 6 , = 2 . 0 ( L p 2 b ) ( 4 8 - 0 1 )
-30.0
--~ ....~_j - ~ ~....-~ ,
10.0 -20.0 0.0
-10.0 =
.
mm -10.0 0
specimens
1
J
I
I
50
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-50
-100
specimens -150
-200
(b)
(a) FIGURE 12. (a) P-~ and (b) P,-~, response for 200-ram specimen corresponding to load-path 2 experiments.
P (kN) 20
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5/"8~ = ~5,,/6s = ~,./6~ =
10
- 10
load-path 2a(47-02) (a)
I
0
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:-0
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i
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200x200x50
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-50
-100
"1,50
-~00
200x200x50
FIGURE 13. Crack patterns for the proportional path (loadpath 2) experiments of Figure 12.
FIGURE 14. (a) P-g and (b) Ps-g, response for the 200-ram specimen corresponding to the simulation of load-path 2.
Advn Cem Bas Mat 1993;1:22-37
Biaxial Tension and Shear on Concrete
lateral deformation ratio (8/8s) = 1.0, whereas the tensile component is more important w h e n 8/8 s = 3.0. This leads to distinct failure modes w h e n the 8/8~ ratio increased from 1.0 to 3.0, as shown in Figure 13. For load-path 2a (8/8, = 1.0), two overlapping straight cracks formed. As compared to load-path I results, the curvature in the overlapping cracks disappeared for load-path 2a. For load-path 2b and c with the deformation ratios 2 and 3, the familiar overlapping cracks were not found (see Figure 13b and c). From the crack pattern it is evident that the tensile mode is only dominant for 8/8s ratios of 2.0 and 3.0. This again reinforces the idea that a higher 8/8~ ratio lowers the influence of the lateral shear. Simulation. The simulated P-8, and Ps-8s plots for load-path 2 with 8/8 s ratios equal to 1.0, 2.0, and 3.0 are shown in Figure 14a and b. The simulated axial tensile peak corresponds to the experimentally obtained values, as can be seen from a comparison of Figures 14a and 12a. Also, the simulated P-8 curves showed the same trend as was observed in the experiment: with increasing 8/8,-ratio the postpeak P-8 curve moved from compression to tension. The main difference with the experiments is the steep falling branches in the simulated P-8 curves. This is due to the brittle nature of the lattice model, as was observed before in the loadpath 1 simulations [11]. The possible explanation for the increased brittleness is the omission of small particles in the simulation as well as the fact that the threedimensional fracture p h e n o m e n o n is treated as a twodimensional problem [12]. The simulated crack pattern
for load-path 2a is the familiar overlapping straight crack mode (Figure 15a). As the deformation ratio increased, the overlapping crack mode disappeared in the simulation too. Again, the simulated crack patterns shown in Figure 15 were found to be identical to the observed crack patterns. INFLUENCE O f SPECIMEN SIZE ON 8/8s = 1.0
Experiment. In these experiments three specimens of three different sizes were used, namely 200, 100, and 50 mm, while the vertical to lateral deformation ratio was kept constant (= 1.0 in this case). In Figure 16a and b, the P/Pmax-8 and Ps-8 s plots are shown for the []11!!~i!~.
(b)
i--
(
(a) FIGURE 15. Simulated crack patterns for the 200-mm specimen with 8/8s = (a) 1.0, (b) 2.0, and (c) 3.0.
33
(c) ['
lliJ!ii~
: ; [ l J [ ; [ l ! ; i ! ! : l i !
34 Nooru-Mohamedet al.
AdvnCem BasMat 1993;1:22-37
P//nmax
2.01.0-
mm mm 1200200110000 mm mm 5O mm mm
~
o.o I
(47-02)
(46-02) {46-07,) (,48-08.) {47- 15) (.48-15)
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I
5O
0
I
IO0
I
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-40.0 -30.0
Ps (kN) o 200mm(47-C x 200mm(46-C [] 100mm(46-C a 100mm(48-C , 50mm(47-1
-20.0
-10.0 mm
0.0
1.0, Lp 20 0
-5O
-1O0
I
-150
-200
FIGURE 16. (a) P/Pm,×-~ and (b) Ps-8, response for load-path 2a experiments on specimens of three different sizes.
load-path 2a tests. The axial load was normalized with respect to the maximum tensile load Pm,× for each individual test. The P~-8~ curves were not normalized because the m a x i m u m shear load P~,m~× was not reached in all tests. As can be seen in Figure 16a, the postpeak P-8 curves shift from tension to compression. A clear indication of the influence of specimen size is seen in this set of curves. As the size increases the steep drop of the normalized P-~ curve is greatly reduced. Each test was duplicated and the results of Figure 16 indicate that an acceptable repeatability was obtained. In spite of the good agreement in P-8 response between the 200-mm tests, two different crack patterns were observed, as shown in Figure 17a and b. A number of reasons for the observed differences are given in ref 10. Without going into too much detail it is mentioned here only that the deviations in cracking may have been caused by the unintentional difference in
notch depth in the two specimens (viz. 20.3 and 26.1 ram, respectively, for specimens 47-02 and 46-02). The crack patterns for the 100- and 50-mm specimens are given in Figure 17c-f. As can be seen in Figure 17c and d, again two identical tests gave two dissimilar crack patterns. From the local deformation measurements of the 100-ram specimen (48-08), the overlapping cracks D and E initiated during the proportional loading regime (8/8s = 1.0). The crack D at the left notch initiated first. However, an identical test with the same 8/8~ ( = 1.0) ratio gave multiple diagonal cracks at the front and the rear faces of the specimen. In contrast, the Ps-8~ curves for both tests were alike (Figure 16b, 100-ram specimen). The crack patterns related to the two 50-mm specimens are given in Figure 17e and f. The familiar overlapping cracks disappear for the 50-mm specimens. Instead, diagonal multiple cracks form at the front and the rear faces of the specimen. For specimen 47-15, the crack F became the dominant crack (front/rear faces), while the other cracks close to a certain extent. Moreover, the dominant crack was tortuous and many fissures appeared on it. It can be concluded that for the ~/~s = 1.0 experiments a combination of failure modes exists, that is, the familiar overlapping crack mode and the inclined distributed crack mode. It is not clear which mode is dominant because in the 200- and 100-ram specimens both modes appeared. Reduction in the specimen size seemed to favor the diagonal multiple cracking mode. In contrast for the load-path lb experiments, all three specimen sizes produced self similar curved overlapping cracks [10]. SPECIMEN SIZE VERSUS 8/8s = 1.0 (LOAD'PATH 2A) Simulation. In this simulation the 8/~s ratio was set to 1. Three specimen sizes were adapted for the mesh. For the 200-mm specimen, a mesh with the random lattice was used. But for the 100-mm specimen, simulations were performed with two different mesh types. They are the random lattice and the regular lattice with embedded aggregate structure. For the 50-mm specimen, the latter mesh only was used. The simulated P/Pm,×-~ and Ps-~s plots in Figure 18a show the tendencies observed in the experiments. The influence of the specimen size is visible in the simulated plots too. As can be seen in Figure 18a, an increase in the specimen size produced a postpeak normalized curve with low compression. As was discussed before, the simulated normalized P-~ curves are too brittle in the postpeak region. The regular lattice and random lattice yield different postpeak behavior. This can be seen, for example, from a comparison of the simulation of the 100-mm specimen with the regular and random lattices, respectively. The postpeak
Biaxial Tension and Shearon Concrete 35
Advn Cem Bas Mat 1993;I :22-37
rear face
front face
1 load-path 2a(47-02) 200x200x50
(a)
rear f a c e ~ (c)
(b)
load-path 2a(46-02) 200x200x50
front face
~.~
"
load-path 2a(46-07) 100xl OOx50 front face
rearface (e)
C '
front face
/
,..,
rear face /
load-path 2a(48-15) 50x50x50
load-path 2a(48-08) 100xl OOx50
(d) front face
rearface load-path 2a(47-15) 50x50x50
(q
FIGURE 17. Crack patterns corresponding to load-path 2a experiments on specimens of three different sizes.
regime of the normalized P-8 curve tends to become less brittle when the regular lattice is used. This is in line with previous analyses which showed, for uniaxial tension, that the brittleness of the simulation is, at least partially, caused by neglected details of the material structure (see ref 12). From Figures 16 and 18 it can be seen that in the load-path 2a simulations, the mesh with the regular lattice and the embedded aggregate structure came closer to the experimental observations. The simulated crack patterns for the four lattices are shown in Figure
19. They are the overlapping straight crack mode for large specimens and changes to the distributed crack mode as the specimen size decreases to 50 mm. Among the two simulated crack patterns for the 100mm specimen, the mesh with the embedded grain structure produced a larger compressive area between the overlapping cracks as compared to the mesh with the random lattice. This probably occurs because the crack path is somewhat governed by the direction of the regular mesh. This is manifested in the simulated normalized P-8 curves too.
36 Nooru-Mohamed et al.
P/F)
Advn Cem Bas Mat 1993;1:22-37
(kN}
max
2
-40 200*200(random) 100.100(.ra nd am) 100*100(regu;ar) 50* 50(regular/
/
-30
/
0 -1
j./-// .j.~'s"" _ /
-lo
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'
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I
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-so
-
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-
-2oo
(a) FIGURE 18. (a)
P/Pmax-~and (b) P~-8 s response for the simulations of load-path 2a for specimens of three different sizes.
Conclusions In this article crack growth under combined tension and shear is investigated by means of experiments and numerical simulations. For this purpose, the recently developed biaxial machine at the Stevin Laboratory was used. The biaxial machine consists of two separate loading frames which can move independently in the axial and the lateral directions.
Two load-paths viz. (1) tension at constant shear, and (2) proportional tension/shear (B/~s = constant) have been investigated. The testing technique is optimized such that any load-path can be followed in a fully automated way. Moreover, the two load-paths have been simulated using a recently developed lattice model. From the experiments it was found that two overlapping cracks formed between the notches w h e n the
(b)
(c) tf !
(d)
(a)
Ilimml~iiilm~liln
FIGURE 19. Simulated crack patterns for the load-path 2a (8/8s = 1.0): (a) 200 m m with random lattice, (b) 100 nun with random lattice, (c) 100 m m with regular lattice with embedded aggregate, and (d) 50 m m with regular lattice with embedded aggregate.
Advn Cem Bas Mat 1993;I :22-37
specimen was subjected first to a lateral shear (mode II) and followed by tension (mode I). A one-to-one relation exists between the lateral compressive shear and the area of the compressive strut which formed between the two overlapping cracks. The axial capacity of the DEN specimens is greatly reduced by the presence of a lateral compressive shear. Self similar overlapping cracks were observed for all three specimen sizes. Thus it seems that the specimen size has no influence on the mode of failure, at least for the specimen ranges covered and pertinent to the load-path pursued. For the experiment with the proportional load-path (8/8s = 1.0), where tension and shear were applied simultaneously, a combination of failure modes exists. When the specimen size was changed from 200 m m to 50 mm, the failure mode changed from the formation of overlapping straight cracks to the growth of a network of inclined distributed cracks. The recently developed lattice model is capable of capturing the curved overlapping cracks found in these load-paths. The simulated peak load corresponds to the experimentally determined value in most cases, using material parameters in the model that were tuned to a uniaxial tensile test. Differences are observed in the postpeak regime. Yet those differences can be explained from the fact that small particles are excluded from the analyses [12] and from the fact that the analyses are carried out in two dimensions whereas the fracture process is in reality a threedimensional phenomenon. It seems that only a mode I criterion is sufficient for predicting crack growth under combined tension and shear.
Appendix. Concrete Composition
and Strength River gravel and sand: 0.125 to 0.25 mm, 303 kg/m3; 0.25 to 0.5 mm, 197 kg/m3; 0.5 to 1.0 mm, 515 kg/m3;
Biaxial Tension and Shear on Concrete
37
1.0 tO 2.0 ram, 500 kg/m3; Portland cement type B, 500 kg/m3; w/c = 0.5; for batch 46, fc = 49.66(2.23) MPa and fspl = 3.76(0.29) MPa; for batch 47, fc = 46.19(0.32) MPa and fspl = 3.78(0.23) MPa; and for batch 48, fc = 46.24(0.37) MPa and f~pl = 3.67(0.29) MPa.
References 1. Wittmann, F.H. Fracture Mechanics of Concrete; Wittmann, F.H., Ed.; Elsevier: Amsterdam, 1983; pp 43-74. 2. Hsu, T.T.C.; Slate, F.O.; Sturman, G.M.; Winter, G. J. Am. Conc. Inst. 1963, 2, 209-224. 3. Van Mier, J.G.M.; Nooru-Mohamed, M.B. Eng. Frac. Mech. 1990, 35, 617-628. 4. Li, B.; Maekawa, K.; Okamura, H. J. Fac. Eng. 1989, 40, 9-52. 5. Saouma, V.; Ayari, M.L.; Boggs, H.L. Fracture of Concrete and Rock; Shah, S.P.; Swartz, S.E., Eds.; Springer Verlag: New York, 1989; pp 311-333. 6. Hassanzadeh, M. Behaviour of Fracture Process Zones in Concrete Influenced by Simultaneously Applied Normal and Shear Displacements; Ph.D. Thesis, Lund Institute of Technology, Lund, Sweden, 1992. 7. Keuser, W.; Walraven, J. Fracture of Concrete and RockRecent Developments; Shah, S.P.; Swartz, S.E.; Barr, B.I.G., Eds.; Elsevier Applied Science Publishers: London, 1988; pp 625--634. 8. Reinhardt, H.W.; Cornelissen, H.A.W.; Hordijk, D.A. Fracture of Concrete and Rock; Shah, S.P.; Swartz, S.E., Eds.; Springer Verlag: New York, 1989; pp 117-130. 9. Van Mier, J.G.M.; Nooru-Mohamed, M.B.; Timmers, G. HERON 1991, 36, 1-104. 10. Nooru-Mohamed, M.B. Mixed-Mode Fracture of Concrete: An Experimental Approach; Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1992. 11. Schlangen, E.; Van Mier, J.G.M. Int. ]. Damage Mech. 1992, 1,435-454. 12. Schlangen, E. Experimental and Numerical Analysis of Fracture Processes in Concrete; Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1993. 13. De Borst, R. Non-Linear Analysis of Frictional Materials; Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1986. 14. Rots, J.G. Computational Modelling of Concrete Fracture; Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1988.
In this article an experimental and numerical study on the behavior of concrete subjected to biaxial loading is outlined. For this purpose the unique biaxial machine available at the Stevin Laboratory was used. Two load-paths were pursued, namely axial tension at constant shear and proportional tension~shear. The recently developed lattice model was used to simulate the two load-paths. The remarkable feature of the lattice model is its ability to simulate curved overlapping cracks which resemble the experimental findings. ADVANCEDCEMENT BASED MATERIALS 1993, 1, 22-37 KEY WORDS: Biaxial loading, Concrete, Fracture, Lattice
model, Load-paths, Shear, Tension, Testing
~
oncrete fracture studies have been pursued in r e c e n t years with great enthusiasm. As a result, the theoretical and experimental knowledge related to mode I fracture of concrete attained its zenith. It was often argued that the real-life problems are seldom confined to mode I fracture alone, and also most structural situations are a combination of tearing (mode I) and shearing (mode II) or twisting (mode III) modes, known as mixed-mode in the classical fracture mechanics terminology. As a natural consequence researchers have attempted to extend the concept of mode I to mixed-mode fracture of concrete. However, to what extent failure is governed by mixed-mode fracture is not straightforward. In this article the results of an extensive study on the behavior of plain concrete subjected to combined tensile (mode I) and in-plane shear loading (mode II) are presented. The intention of most fracture mechanics research of concrete has been the determination of material fracture properties for numerical models in structural analysis. The macroscopic p h e n o m e n o n observed in an ex-
m
Address correspondence to: Jan G.M. van Mier, Stevin Laboratory, Faculty of Civil Engineering, Delft University of Technology, P.O. Box 5048, 2600 6A Delft, The Netherlands. © Elsevier Science Publishing Co., Inc. ISSN 1065-7355/93/56.00
periment is to a large extent due to the processes occurring at a lower level [1]. The presence of aggregative constituents is the main cause of heterogeneity in concrete. Based on the ratio between the maximum aggregate size and the characteristic dimension of a specimen, the structure (specimen) can be treated either as homogeneous or as heterogeneous. At the plastic and hardened stage, nonstructural intrinsic cracks form due to the nature of the various constituents of concrete. Therefore, cracks exist at the aggregate mortar interface even before any mechanical load is applied [2]. Two regions, prepeak and postpeak, can be identiffed on a macroscopic global load-deformation (P-8) curve of a uniaxial tensile bar. In the prepeak region micro cracks evolve, but in comparison to the heterogeneity of the material, these cracks are relatively small and w h e n a representative volume is considered, the continuum theory is valid in this region. However, in the postpeak region, conveniently referred to as softening region, discrete cracks form. Here, the size of the discrete crack is of the same order of magnitude as that of the specimen, and the continuum theory is violated. The final global response of the experiment is a combination of the response of the material, the geometry of the specimen, and the boundary conditions under which the test is carried out. It is important to distinguish between the material and structural properties in a (fracture) test. However, these differences cannot be identified in a straightforward manner. It is doubtful whether one can ever obtain the real material fracture properties without the various influences, such as the boundary conditions, the specimen shape/size, etc [3]. The capacity of a partially opened crack to transfer shear is of practical interest. The curved crack which forms in reinforced concrete beams or columns [4] and cracks growing in the upstream face of a gravity dam [5] are two such examples. When the crack openings are small, the transfer of shear is not only associated with aggregate interlock mechanism but is also related Received February 9, 1993 Accepted May 25, 1993
Advn Cem Bas Mat 1993;I :22-37
Biaxial Tension and Shear on Concrete
to fracturing of the aggregate mortar matrix. Attempts were made in refs 6 and 7 to link the aggregate interlock theories for cracks loaded in simultaneous tension and shear. The existing aggregate interlock models fall short of capturing the actual phenomenon when the crack openings are small. The second effect, fracturing of the aggregate mortar matrix, is normally neglected. The purpose of the present fracture studies is to broaden the knowledge to the situation where tension and shear act simultaneously. The unique biaxial machine [8-10] built at the Stevin Laboratory of Delft University of Technology was used. The machine consists of two independent frames which are capable of moving along the vertical and horizontal directions. Axial load and lateral shear are applied to the specimen through this configuration of independent loading frames. Double-edge-notched (DEN) specimens of three different sizes are used. Experimental research and numerical tools must play a complementary role to arrive at a better understanding of the behavior of concrete. On the one hand numerical tools can be utilized to interpret the experi-
mental observations, and on the other hand experimental results can be used to tune a numerical model. Because the fracture process is a discrete phenomenon, the continuum-based approaches fall short of adequately describing this behavior. Therefore, a more convincing approach might be the use of micromechanics-based models to study fracture processes. In this article the experimental observations of two load-paths are discussed, namely axial tension at constant shear (load-path 1) and proportional tension/ shear (load-path 2). Formation of curved overlapping cracks is the main feature of load-path 1. The recently developed lattice model [11,12] was used to simulate load-paths 1 and 2. The remarkable feature of the lattice model is its ability to simulate curved overlapping cracks which resemble exactly the experimental findings. Further comparison is made between results of the numerical simulation and the experimental loaddisplacement curves. It seems that the mode I criterion adopted in the lattice model is sufficient for simulating the biaxial tension/shear behavior observed in the experiments.
security ~ device IJ~ll~
coupled
.i ' "
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. II
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-
.
\ food-
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24 Nooru-Mohamed et al.
Advn Cem Bas Mat 1993;1:22-37
Testing Technique In this section a summary of the experimental technique is given. For a more extensive description of the biaxial testing machine and the test technique, refer to ref 10.
Materials and Specimens In this study, DEN square concrete elements of three different sizes were used; namely 200 x 200, 100 x 100, and 50 x 50 mm, with equal thickness of 50 m m (each having a constant notch to depth ratio a/d = 0.125). The maximum aggregate size was 2 mm. Details of the concrete mix are given in the Appendix.
Loading Frame To study the behavior of concrete under biaxial tension/shear loading, a unique machine was developed at the Stevin Laboratory of Delft University of Technology. The biaxial machine consists of two independent, stiff, square loading frames. The exploded view of the machine is shown in Figure 1. The outer frame A is a coupled frame and it is capable of moving along the horizontal direction. The inner frame B can move along the vertical direction. Moreover, frames A and B can move independently of one another. The outer frames and the inner frame are fixed to the overall frame via eight vertical and four horizontal plate springs. The assumption is that the plate springs pre(a)
plote springs
(b)
FIGURE 2. Photograph of the biaxial test set-up. vent the frames from rotating. They have a limited bending capacity. On each side of the plate spring two strain gauges are mounted and connected to the data acquisition system. Any eccentricity which may occur during a test in the frame assembly could be traced back from the plate spring readings. Two hydraulic actuators (A,B) and two load cells (A,B) are situated outside the loading frames so as not to hinder the overall stiffness of the loading frame. The frame assembly is shown in Figure 2. In Figure 3a and b, the sectional view of the loading middle frome
bending of plote springs
connecting plote for coupled fromes
FIGURE 3. Sectional view of the biaxial loading frames in two stages: (left) before and (right) after the specimen is cracked.
Advn Cem Bas Mat 1993;1:22-37
Biaxial Tension and Shear on Concrete
frame
either the average vertical deformation ~ or the average global shear deformation 8s was used as the feedback signal in the closed loop servo control system.
B
Measurements and Data Acquisition
10
c~ 10
us
25
~
14o
FIGURE 4. Boundary condition for a DEN specimen, subjected to combined tension and shear. frame is shown in two stages. Here a DEN specimen is glued to the frames. As can be seen from a and b, the top of the specimen is glued to the inner frame and the bottom of the specimen to the coupled outer frame. The inner frame can slide in between the coupled frame, as shown in a. The specimen is loaded by moving the middle frame upward. A tensile stress develops in the concrete specimen because the coupled outer frames are fixed in the vertical direction. A tensile crack always nucleates in the notched region, which is located at half height of the DEN specimen (Figure 3b), so that crack evolution can be followed at a known location. This enables deformation controlled testing. Note the b e n d i n g of the horizontal plate springs in Figure 3b after crack formation in the DEN specimen. The DEN specimens are fixed to the frames via loading platens. For a uniaxial tensile test only two loading platens are required. However, for a biaxial test two additional loading platens are needed to apply a lateral in-plane shear load. The assumed boundary condition for a DEN specimen is outlined schematically in Figure 4. The boundary conditions allow for parallel end-plate movement in the vertical and horizontal directions during the test.
Test Control The mixed-mode machine has two independent frames connected to two independent regulation circuits, and it is therefore possible to control the vertical and horizontal deformations separately in a test. Moreover, tests can also be performed with deformation control in the vertical direction and load control in the horizontal direction and vice versa. The average signal of the control LVDTs was amplified via a standard amplifier (SCHENCK MV 318) to the servo controller unit. A function generator was connected to the servo controller. Depending on the type of test, the signals of
The signals related to the movement of the two loading frames A (the outer frame, capable of moving in the horizontal direction) and B (the inner frame, which can move in the vertical direction) were connected to two independent regulation circuits 1 and 2 of the SCHENCK regulation system. The machine contains two actuators and two load-cells (see Figure 1). The capacity of the loading axes is restricted to 50 kN in the vertical and horizontal directions because of the limited strength of the plate springs. For the 200- and 100-ram specimens, four LVDTs (Sangamo A6G1, gauge length = 65 ram), and for the 50-ram specimen two LVDTs (gauge length = 35 ram) were used. The LVDTs were calibrated using a device containing a Mitutoyo micro-screw with 0-50-ram movement and a 5-~m accuracy. The LVDT has a linear stroke of --1 m m with a nonlinearity of 0.3%. For the lateral d e f o r m a t i o n m e a s u r e m e n t s two LVDTs (Sangamo, A6G1) were used in the horizontal direction. At the front face of the 200-ram specimen, one LVDT was mounted at the center of the bottom half, as shown in Figure 4, the second LVDT was fixed to the diagonally opposite position at the rear face of the specimen. The global shear displacement ~s was defined as the average displacement between the specimen halves measured by the two LVDTs. In addition to the LVDTs, a number of locally built extensometers were used to measure local deformations. The extensometer has a range of 0 to 2 ram, a resolution of 1 p,m, a nonlinearity of 0.5%, and an accuracy of 5 p,m. The same calibration device mentioned above was used to calibrate the extensometers. The number of extensometers used varied depending on the type of test and the size of the specimen. A 32 channel programmable data logger (designed by the Measurement and Instrumentation Group of the Stevin Laboratory) was used to retrieve the data from the extensometers and strain gauges. The total scan t~'ne for 32 channels is 976 ms. A micro computer was required to program the data logger. Dedicated software was developed that allowed fully automatic testing. The personal computer controls the data logger, the SCHENCK regulation system, the A/D convertor, and the function generator. During a test scanning was done at fixed time intervals. The load was applied continuously and no stops were made for scanning of the measuring devices. The data logger was programmed to scan every 10 seconds from the beginning of a test, with the capability of changing the scan time (usually 30 to 60 seconds). A
26
Nooru-Mohamed et al.
Advn Cem Bas Mat 1993;1:22-37
typical scan consists of signals from two loads (P, P~) [mV], two deformations (6,5s) [mV], six to 20 extensometers [I~V], and 12 strain gauges [txV] from the vertical and horizontal plate springs.
Results and Discussions Load-Path Description In this article the direction of the lateral shear load is shown in Figure 4. When referring to the crack opening, it will be implied that no correction was made for the linear elastic deformation within the measuring length of the LVDT or extensometer. In Figure 5, the two load-paths investigated are shown schematically as P-P~, P-6, and 6-time diagrams. They are the following: LOAD-PATH 1
Axial Tension at a Constant Shear Force (Figure 5a). A compressive shear load Ps was applied to the specimen in displacement control until Ps = - 5 kN, - 1 0 kN,
and to the maximum shear load the specimen could sustain for load-path la, lb, and lc, respectively. LOAD-PATH 2
Proportional Loading (Figure 5b). An axial tensile and a lateral compressive shear load were applied to the specimen such that the ratio of axial (5) to lateral deformation (Ss) remained constant throughout the test. The 6/Ss-ratio for load-path 2a, 2b, and 2c was 1.0, 2.0, and 3.0, respectively.
Experimental Results and Numerical Simulations INFLUENCE OF LATERAL CONFINEMENT (LOAD-PATH 1)
Experiments. In the load-path 1 experiments, first a lateral compressive shear was applied in displacement control. Subsequently, the lateral test control was changed to load control and an axial tensile load was applied in displacement control. In other words, a constant lateral confinement was maintained while the ax-
Ps = -5,-lO, P~,mo
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I=
P
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FIGURE5. Schematic representation of load-paths I and 2. In (a) load-path 1, first shear is applied (P, = -5, - 10, and maximum shear in load-path la, b, and c, respectively), followed by axial tension at constant P,; In (b) load-path 2 proportional 8/8~ is applied.
Advn Cem Bas Mat 1993;1:22-37
Biaxial Tension and Shearon Concrete 27
(kN) 0 []
P , = - 5 kN ILp a ) ( 4 8 - 0 3 ) P , = - I O kN Lp I b ) ( 4 6 - 0 5 ) P , = - I O kN fLP 11b)(47-01 ) Ps= max Lp c ) ( 4 7 - 0 6 )
a
-40.0
*
-30.0
/••o
:'2 mm 200 mm specimens
-20.0 -10.0
.m".. 0
0.0
~
I
- 50
the 8i-~s plot (Figure 7a), the largest increase in 8i first occurred at gauges 1 and 2 and subsequently at gauge 5. In Figure 7b, the locations of the measuring points are identified. This confirms the crack initiation at the left and right notches. The crack opening was faster at the left notch than at the right notch. Evidently, this crack initiation occurred during subsequent axial pulling at a constant lateral compressive shear (Ps) equal to - 1 0 kN. The two overlapping curved cracks EF and GH with a compressive strut (i.e., the enclosed area between EF and GH) in the middle were formed at the end of the test, as shown in Figure 8b. Specimen 48-03 subjected to load-path la behaved in a similar fashion. However, the overlapping cracks AB and CD (Figure 8a) were more fiat, which resulted in a smaller strut size. For the specimen 47-06 which was first loaded up to maximum shear, the Ps-Bs diagram became nonlinear at P~ = - 12.9 kN and Bs = - 30 I~m (Figure 6a). This nonlinearity is due to the crack initiation at the front in
~*
I
I
-100
-200
-150
(a)
P (kN) 20.0 o n n~ a ,k*
15.0 j
P , = - 5 kN p=-lO kN P , = - I O kN P'= max
10.0 - ~ \
(Lp (Lp (~Lp (Lp
lo)(48-03) lb)(46-05) lb)(47-01) lc)(47-06)
do -- 2 mm
(~i
5O
o gouge [] gauge a gouge 0 gauge , gauge specimen load-path
45 4O
.
35 5.0
3O
--
25
A
0.0
1 2 3 4 5 46-05 lb
20
-5.0
0
r
i
,
50
100
150
15200
105-
FICURE 6 . (a) Ps-$s and (b) P-8 response for 200-mm specimen corresponding to load-path 1 experiments.
0 -5
ial tensile force was activated. In Figure 6a and b, the P,-B, and the P-B diagrams for the 200-mm specimens corresponding to load-path la, b, and c are shown. The shear load for load-path la, b, and c was - 5 , - 1 0 , and -27.5 kN (maximum shear), respectively. For specimen 46-05 (Figure 6a), the Ps-8~ plot is linear until P~ = - 10 kN. At P, = - 10 kN, the shear deformation, 8s, and the highest local deformation, Bi (Figure 7), corresponding to the measuring point 1 were - 2 2 I~m and 4 I~m, respectively. At point B on the Ps-~s diagram where ~ = - 2 2 I~m for load-path lb (Figure 6a), the tensile load is activated. As can be seen
I
0
-10
I
-20
I
-30
I
-40
I
-50
(a)
(b) F I C U R E 7. (a) Local deformation versus lateral shear (8i-8~) and the (b) location of measuring points for specimen 46-05 (load-path lb).
2 8 Nooru-Mohamed et
al.
Advn Cem Bas Mat 1993;1:22-37
front face
(a) rear face
load-path la(48-03) front face
(b) rear face
load-path lb(46-05) front face
L
(c)
load-path lc(47-06)
FIGURE 8. Crack patterns for the lateral confinement experiments (load-path 1) of Figure 6. left notch (measuring point 1), as can be seen in the 8i-8s plot in Figure 9a. Further shearing led to a maxim u m shear load of - 27.5 kN, at which time the shear deformation was - 9 2 . 4 p,m. As can be seen in Figure 9, the largest increase in 8 i occurred first at gauge 1, subsequently at gauges 11, 6 (811 = 86), and 7. Gauges 1 and 11 were located near the left notch at the front and rear faces of the specimen. Evidently, the crack propagated on the front face of the specimen from the left notch to the location of gauge 7. This crack extension is identified as IJ on the crack pattern shown in Figures 8c and 9c. However, on the rear face the fracture propagation clearly lagged behind during the application of the lateral shear load. The crack on the rear face reached gauge 17 (located opposite to gauge 7) only after activating the axial tension. The crack prop-
agation during the lateral shear caused an increase of (note that the LVDTs were mounted near the notches). This can clearly be seen from Figure 6b: point A on the P-~ diagram for load-path lc corresponds to the axial deformation during the application of the lateral compressive shear load. (In the shear regime, load-control was maintained in the axial direction and P = 0 = constant.) At A the axial tensile loading was activated, maintaining a constant lateral shear load of - 2 7 . 5 kN (slightly below the maximum shear capacity of the specimen). As can be seen in Figure 6b, the axial load P became compressive almost immediately after it was activated. The crack KL initiated at the right notch. Thus, during crack propagation a small axial compressive load had to be maintained to obtain a stable test result. The highest axial tensile load (15 kN) was measured in the test with the low shear load of - 5 kN. A 30% decrease in P occurred when Ps was increased from - 5 kN to - 1 0 kN. Clearly the presence of a lateral compressive shear load (or a lateral confinement) significantly affected the axial tensile capacity of the specimen. Figure 8 shows the crack patterns for the loadpath 1 experiments. A relation seems to exist between the lateral shear load (or lateral confinement) and the area of the compressive struts. The specimen with the maximum shear load (Figure 8c) shows the largest strut area, whereas the specimen with the lowest shear (Figure 8a) gives the smallest strut size. In all the specimens the crack initiated near the notch at an angle of approximately 45 °. Owing to the application of the lateral compressive shear and the axial tension, the stress state changes at the crack tip and the principal stresses rotate. Due to this principal stress rotation, the crack propagates along a curvilinear path. To check the repeatability of the test results a small number of tests were duplicated in a random manner. As can be seen in Figure 6b, two tests of Ps = - 10 kN were repeated. There exist some deviations in the prepeak behavior of the two specimens that can likely be attributed to the difference in the age of loading (48 and 157 days, respectively). However, self similar crack patterns were observed for both specimens. Numerical simulation of fracture processes. Numerical tools can be used to gain some insight into the fracture behavior of concrete structures. Experimental research and numerical tools must play a complementary role to arrive at a better understanding of the behavior of complex disordered materials like concrete. On the one hand numerical tools can be utilized to interpret the experimental observations, and on the other hand experimental results can be used to tune a numerical model. The fracture process in concrete is a discrete phenomenon. The continuum-based approaches fall short
Advn Cern Bas Mat 1993;1:22-37
o a ,, 0
125
gauge gauge gauge gauge
Biaxial Tension and Shear on Concrete
1 6 7 5
front; front front; front;
, gauge 11 rear x gouge 17 rear . gouge 15 rear
specimen 4 7 - 0 6 load-path 1c
•:11Bi:
75 I
.s
/
.~-"'"
25
-25
29
//t
/
lIi +: +...
..... ..... ...
ili 2"ii .....
6 " ....
+:l0:, ®.++ o;2O..:, 1 """ • ....
9 "-
3
~1131! .....
®
:1~4 4 " ..... '
)
7
)
,iiiiiii rear face I
0
I
-5
b
'
I
-1 O0
'
l
- 150
(b)
(a)
g o u g e 17 (c)
gouge 7
FIGURE 9. (a) 8i-8s, (b) measuring points, and (c) crack evolution during the lateral shear, for load-path lc experiments.
of capturing or adequately describing this behavior. The formation of a macro crack depends to a large extent on processes occurring at a lower (micro-mesa) level of the material. Therefore, a more convincing approach might be the use of micro-mechanics-based models to study fracture processes. The recently proposed lattice model (for details, refer to refs 11 and 12) is used to simulate the two load-paths (1 and 2) discussed in this article. In the following paragraphs, a comparison is made between the numerical simulation and the experimental results. In the lattice model the material is schematized as a lattice of brittle breaking beam elements. A triangular lattice is chosen for convenience. Three options for intraducing disorder in the lattice model have been proposed thus far. In the first approach, strength and/or stiffness values drawn from a statistical distribution are assigned to the individual beam elements in a regular lattice. Second, the lattice can be overlaid on top of a g e n e r a t e d grain s t r u c t u r e of concrete. Different strength and stiffness values are assigned to beam el-
ements overlapping with the aggregate particles, the matrix material, and the bond zone. Third, a random lattice with a variable beam length can be used with a constant beam strength. A simple fracture law is implemented in the lattice model. Fracture is simulated by removing in each loadstep the element whose stress has exceeded the assigned tensile strength. A linear finite element analysis is only required to evaluate the stress, and at each load step the above procedure is repeated. It seems that softening can be omitted altogether in this approach. However, a parameter study is required before simulating various load-paths. Accordingly, the relevant parameters of the lattice model were tuned based on the uniaxial tensile experiments [12]. Note that only a small number of single valued parameters suffice for describing the local material properties. SIMULATION OF LOAD-PATH 1 Tension at a Constant Shear. In Figure 10a and b, the
simulated Ps-Ss and P-8 plots for load-path 1 are shown. The simulations were stopped at 8s ~ 60 ~m. According to the load-path applied in the experiments, the shear loads were kept constant at - 5 kN and - 10 kN for path la and lb, respectively. The simulated crack patterns are shown in Figure 11. For the loadpath 1 simulations, one specimen size (viz. 200 mm) was used. The lattice model with a variable beam length was adopted in these analyses. In line with the observations, the formation of the
30 Nooru-Mohamed et al.
Advn Cem Bas Mat 1993;1:22-37
Ps (kN} Lp Lp Lp
-40 -
~q :b ~c,
-30
-20
-10
/
0
'
[
I
T
I
1
-150
-10,.-
-50
-2CC
I
(a)
P (kN} 20
vf t/I
15 ~P
10
Lp
I
lb
I ',+, I /
i
/
j /
/I,,.
I
'~ . " "i
iI
~
L
I
0
-5
i
0
I
50
I
I
100
i
I
150
r
200
FIGURE 10. (a) Ps-~s and (b) P-8 response for the 200-mm specimen corresponding to the simulation of load-path 1. curved overlapping cracks was found in the simulations. The remarkable feature of the lattice model is that it can capture the curved cracks. Contrary to this, continuum-based approaches such as the smeared crack model fail to produce the curved cracks; also, the crack often forms along the boundaries of the element mesh [13,14]. A (qualitative) comparison between the simulated and the experimental crack patterns shows a close resemblance (compare Figures 8 and 11). An in-
crease in the lateral confinement produced an increase in the confined area between the overlapping cracks. The same observation was made in the experiment too (see Figure 8). For load-path la, the simulated Ps-~s diagram exactly matches the experimental result. Also, the simulated peak shear load (load-path lc) is identical to that of the experimental value. However, for load-path lb and c with Ps = - 10 kN and maximum shear, respectively, the simulated Ps-~s diagrams show a higher stiffness as compared to the experimental results (see Figures 10a, and 6a). This is caused by the additional plane stress elements attached to the halves of the mesh edges A and B (see Figure 11a and b). These additional plane stress elements with an equivalent concrete stiffness were provided for the simulations with Ps = - 10 kN and the maximum shear only. The addition of stiff elements led to a higher shear load, and therefore a visible higher stiffness of the initial part of the P~-~ diagram. The purpose of adding these stiff elements was to prevent premature cracking along the specimen loading interface. However, this goal was only partially achieved because, at the peak shear load, cracks still formed along the boundary. To avoid the formation of boundary cracks at the peak shear load, simulations were repeated for load-path lc with an additional column of plane stress elements with a stiffness equivalent to that of steel. The influence of the loading platen stiffness on crack initiation can clearly be seen in Figure 11c and d. Both simulated crack patterns correspond to the situation at maximum compressive shear load. For the mesh with lower stiffness (Figure 11c), in addition to the familiar overlapping cracks, boundary cracks as well as strut-area cracks formed. However, for the case with a higher loading platen stiffness (in fact resembling the well-known shear box experiments), the additional cracks near the loading platen did not appear (see Figure 11d). This indicates that a shear box type loading reduces stress concentrations at the two corners (top right/bottom left) of the specimen. In that case, the formation of glue layer cracks is unlikely w h e n the maximum shear force is applied. In the experiments the occurrence of additional boundary cracks was not seen for large specimen sizes, namely 200 ram. However, the above phenomenon was frequent in the case of specimens with a thickness of 25 m m instead of 50 mm. It seems that the out-ofplane stiffness is not adequate for a stable test. Moreover, the boundary cracks were also prevalent w h e n small specimens (viz. 100 and 50 m m with normal thickness of 50 ram) were used. The individual plate spring reading would show some deviations if there were glue layer cracks. Hence, the vertical load transfer would not be uniform.
Biaxial Tension and Shear on Concrete 31
Advn Cem Bas Mat 1993;1:22-37
A
B
(b)
(a)
(c)
V I I I I I I I I I I J I I I I I I I I I I ) I [ I I J I I I I I I I I I ] I
(d)
I I I l i l l l l I ] l l l [ l l l l ] J l l l l t
jl~r[zr]=~iz~l
j-r-~, ~I I I I l l l l t l i l ~l l l ] l l Il I [ l l l l l l l l l [ l l [ l l rl - c a a c a P ~ , ,~
FIGURE 11. Simulated crack patterns with random lattice: (a) Ps = - 5 kN, (b) Ps = - 1 0 kN with additional column of plane stress elements at edges A and B, (c) Ps = maximum shear with mesh as in (b), and (d) Ps = maximum shear with two columns of plane stress elements with high stiffness.
Proportional Loading (Load-Path 2) INFLUENCE OF THE 8/8 s RATIO FOR THE 200 M M SPECIMEN
Experiment. In load-path 2, the axial tensile a n d the lateral shear load w e r e applied such that the 8/8s ratios were constant a n d equal to 1.0, 2.0, a n d 3.0. The spec-
imen size was 200 m m . Figure 12a a n d b s h o w s the P-8 a n d Ps-Ss plots for the 200-mm specimen. The influence of the 8/8s ratio is seen in these plots. For 8/8s = 1.0, the p o s t p e a k P-8 curve w e n t into c o m p r e s s i o n as c o m p a r e d to the o t h e r two ratios. Moreover, the 8/8~ = 1.0 p r o d u c e d the highest lateral shear. The influence of the lateral shear is m o r e d o m i n a n t w h e n the axial-
32 Nooru-Mohamed et al.
Advn Cem Bas Mat 1993;1:22-37
P (kN) o [] z~
ps (kn)
-40.0
20.0
6/8,=I.0 (Lp 20)(47-02) 5Z6~=I.0 (Lp 2a)(46-02~ 6~6,--210 {,Lp 2b){48-01)
:::)6/6s= I .0(Lp20)(47-02) u 6Z8.= 1.0(Lp2a)(46-02.) ,, 6 Z 6 , = 2 . 0 ( L p 2 b ) ( 4 8 - 0 1 )
-30.0
--~ ....~_j - ~ ~....-~ ,
10.0 -20.0 0.0
-10.0 =
.
mm -10.0 0
specimens
1
J
I
I
50
t O0
150
200 mm
F
0.0 0
200
-50
-100
specimens -150
-200
(b)
(a) FIGURE 12. (a) P-~ and (b) P,-~, response for 200-ram specimen corresponding to load-path 2 experiments.
P (kN) 20
rear face
5/"8~ = ~5,,/6s = ~,./6~ =
10
- 10
load-path 2a(47-02) (a)
I
0
200x200xS0
I
front face
[
:-0
[
I O0
1.0 2.0 3.0
I
150
200
i
P~ (kN) -40
-5o
[
es/~,
=
2.0
load-path ~48-01) (b)
200x200x50
i -20
//
... -
front face
0
(c)
-50
-100
"1,50
-~00
200x200x50
FIGURE 13. Crack patterns for the proportional path (loadpath 2) experiments of Figure 12.
FIGURE 14. (a) P-g and (b) Ps-g, response for the 200-ram specimen corresponding to the simulation of load-path 2.
Advn Cem Bas Mat 1993;1:22-37
Biaxial Tension and Shear on Concrete
lateral deformation ratio (8/8s) = 1.0, whereas the tensile component is more important w h e n 8/8 s = 3.0. This leads to distinct failure modes w h e n the 8/8~ ratio increased from 1.0 to 3.0, as shown in Figure 13. For load-path 2a (8/8, = 1.0), two overlapping straight cracks formed. As compared to load-path I results, the curvature in the overlapping cracks disappeared for load-path 2a. For load-path 2b and c with the deformation ratios 2 and 3, the familiar overlapping cracks were not found (see Figure 13b and c). From the crack pattern it is evident that the tensile mode is only dominant for 8/8s ratios of 2.0 and 3.0. This again reinforces the idea that a higher 8/8~ ratio lowers the influence of the lateral shear. Simulation. The simulated P-8, and Ps-8s plots for load-path 2 with 8/8 s ratios equal to 1.0, 2.0, and 3.0 are shown in Figure 14a and b. The simulated axial tensile peak corresponds to the experimentally obtained values, as can be seen from a comparison of Figures 14a and 12a. Also, the simulated P-8 curves showed the same trend as was observed in the experiment: with increasing 8/8,-ratio the postpeak P-8 curve moved from compression to tension. The main difference with the experiments is the steep falling branches in the simulated P-8 curves. This is due to the brittle nature of the lattice model, as was observed before in the loadpath 1 simulations [11]. The possible explanation for the increased brittleness is the omission of small particles in the simulation as well as the fact that the threedimensional fracture p h e n o m e n o n is treated as a twodimensional problem [12]. The simulated crack pattern
for load-path 2a is the familiar overlapping straight crack mode (Figure 15a). As the deformation ratio increased, the overlapping crack mode disappeared in the simulation too. Again, the simulated crack patterns shown in Figure 15 were found to be identical to the observed crack patterns. INFLUENCE O f SPECIMEN SIZE ON 8/8s = 1.0
Experiment. In these experiments three specimens of three different sizes were used, namely 200, 100, and 50 mm, while the vertical to lateral deformation ratio was kept constant (= 1.0 in this case). In Figure 16a and b, the P/Pmax-8 and Ps-8 s plots are shown for the []11!!~i!~.
(b)
i--
(
(a) FIGURE 15. Simulated crack patterns for the 200-mm specimen with 8/8s = (a) 1.0, (b) 2.0, and (c) 3.0.
33
(c) ['
lliJ!ii~
: ; [ l J [ ; [ l ! ; i ! ! : l i !
34 Nooru-Mohamedet al.
AdvnCem BasMat 1993;1:22-37
P//nmax
2.01.0-
mm mm 1200200110000 mm mm 5O mm mm
~
o.o I
(47-02)
(46-02) {46-07,) (,48-08.) {47- 15) (.48-15)
-1.0 do
~---
eS/G = 1.0, Lp 2a -2.0
I
5O
0
I
IO0
I
150
2OO
(a)
-40.0 -30.0
Ps (kN) o 200mm(47-C x 200mm(46-C [] 100mm(46-C a 100mm(48-C , 50mm(47-1
-20.0
-10.0 mm
0.0
1.0, Lp 20 0
-5O
-1O0
I
-150
-200
FIGURE 16. (a) P/Pm,×-~ and (b) Ps-8, response for load-path 2a experiments on specimens of three different sizes.
load-path 2a tests. The axial load was normalized with respect to the maximum tensile load Pm,× for each individual test. The P~-8~ curves were not normalized because the m a x i m u m shear load P~,m~× was not reached in all tests. As can be seen in Figure 16a, the postpeak P-8 curves shift from tension to compression. A clear indication of the influence of specimen size is seen in this set of curves. As the size increases the steep drop of the normalized P-~ curve is greatly reduced. Each test was duplicated and the results of Figure 16 indicate that an acceptable repeatability was obtained. In spite of the good agreement in P-8 response between the 200-mm tests, two different crack patterns were observed, as shown in Figure 17a and b. A number of reasons for the observed differences are given in ref 10. Without going into too much detail it is mentioned here only that the deviations in cracking may have been caused by the unintentional difference in
notch depth in the two specimens (viz. 20.3 and 26.1 ram, respectively, for specimens 47-02 and 46-02). The crack patterns for the 100- and 50-mm specimens are given in Figure 17c-f. As can be seen in Figure 17c and d, again two identical tests gave two dissimilar crack patterns. From the local deformation measurements of the 100-ram specimen (48-08), the overlapping cracks D and E initiated during the proportional loading regime (8/8s = 1.0). The crack D at the left notch initiated first. However, an identical test with the same 8/8~ ( = 1.0) ratio gave multiple diagonal cracks at the front and the rear faces of the specimen. In contrast, the Ps-8~ curves for both tests were alike (Figure 16b, 100-ram specimen). The crack patterns related to the two 50-mm specimens are given in Figure 17e and f. The familiar overlapping cracks disappear for the 50-mm specimens. Instead, diagonal multiple cracks form at the front and the rear faces of the specimen. For specimen 47-15, the crack F became the dominant crack (front/rear faces), while the other cracks close to a certain extent. Moreover, the dominant crack was tortuous and many fissures appeared on it. It can be concluded that for the ~/~s = 1.0 experiments a combination of failure modes exists, that is, the familiar overlapping crack mode and the inclined distributed crack mode. It is not clear which mode is dominant because in the 200- and 100-ram specimens both modes appeared. Reduction in the specimen size seemed to favor the diagonal multiple cracking mode. In contrast for the load-path lb experiments, all three specimen sizes produced self similar curved overlapping cracks [10]. SPECIMEN SIZE VERSUS 8/8s = 1.0 (LOAD'PATH 2A) Simulation. In this simulation the 8/~s ratio was set to 1. Three specimen sizes were adapted for the mesh. For the 200-mm specimen, a mesh with the random lattice was used. But for the 100-mm specimen, simulations were performed with two different mesh types. They are the random lattice and the regular lattice with embedded aggregate structure. For the 50-mm specimen, the latter mesh only was used. The simulated P/Pm,×-~ and Ps-~s plots in Figure 18a show the tendencies observed in the experiments. The influence of the specimen size is visible in the simulated plots too. As can be seen in Figure 18a, an increase in the specimen size produced a postpeak normalized curve with low compression. As was discussed before, the simulated normalized P-~ curves are too brittle in the postpeak region. The regular lattice and random lattice yield different postpeak behavior. This can be seen, for example, from a comparison of the simulation of the 100-mm specimen with the regular and random lattices, respectively. The postpeak
Biaxial Tension and Shearon Concrete 35
Advn Cem Bas Mat 1993;I :22-37
rear face
front face
1 load-path 2a(47-02) 200x200x50
(a)
rear f a c e ~ (c)
(b)
load-path 2a(46-02) 200x200x50
front face
~.~
"
load-path 2a(46-07) 100xl OOx50 front face
rearface (e)
C '
front face
/
,..,
rear face /
load-path 2a(48-15) 50x50x50
load-path 2a(48-08) 100xl OOx50
(d) front face
rearface load-path 2a(47-15) 50x50x50
(q
FIGURE 17. Crack patterns corresponding to load-path 2a experiments on specimens of three different sizes.
regime of the normalized P-8 curve tends to become less brittle when the regular lattice is used. This is in line with previous analyses which showed, for uniaxial tension, that the brittleness of the simulation is, at least partially, caused by neglected details of the material structure (see ref 12). From Figures 16 and 18 it can be seen that in the load-path 2a simulations, the mesh with the regular lattice and the embedded aggregate structure came closer to the experimental observations. The simulated crack patterns for the four lattices are shown in Figure
19. They are the overlapping straight crack mode for large specimens and changes to the distributed crack mode as the specimen size decreases to 50 mm. Among the two simulated crack patterns for the 100mm specimen, the mesh with the embedded grain structure produced a larger compressive area between the overlapping cracks as compared to the mesh with the random lattice. This probably occurs because the crack path is somewhat governed by the direction of the regular mesh. This is manifested in the simulated normalized P-8 curves too.
36 Nooru-Mohamed et al.
P/F)
Advn Cem Bas Mat 1993;1:22-37
(kN}
max
2
-40 200*200(random) 100.100(.ra nd am) 100*100(regu;ar) 50* 50(regular/
/
-30
/
0 -1
j./-// .j.~'s"" _ /
-lo
.z~/~-" "-~ 0 />'f, ,
-2
'
0
I
50
'
I
100
'
I
150
tOO,2OOlrandom~l
"
oo,100,random/I
• /
100, 100(regatar) ] , 5,0* bO{regular)
'
,..}0
o
-so
-
oo
-
-2oo
(a) FIGURE 18. (a)
P/Pmax-~and (b) P~-8 s response for the simulations of load-path 2a for specimens of three different sizes.
Conclusions In this article crack growth under combined tension and shear is investigated by means of experiments and numerical simulations. For this purpose, the recently developed biaxial machine at the Stevin Laboratory was used. The biaxial machine consists of two separate loading frames which can move independently in the axial and the lateral directions.
Two load-paths viz. (1) tension at constant shear, and (2) proportional tension/shear (B/~s = constant) have been investigated. The testing technique is optimized such that any load-path can be followed in a fully automated way. Moreover, the two load-paths have been simulated using a recently developed lattice model. From the experiments it was found that two overlapping cracks formed between the notches w h e n the
(b)
(c) tf !
(d)
(a)
Ilimml~iiilm~liln
FIGURE 19. Simulated crack patterns for the load-path 2a (8/8s = 1.0): (a) 200 m m with random lattice, (b) 100 nun with random lattice, (c) 100 m m with regular lattice with embedded aggregate, and (d) 50 m m with regular lattice with embedded aggregate.
Advn Cem Bas Mat 1993;I :22-37
specimen was subjected first to a lateral shear (mode II) and followed by tension (mode I). A one-to-one relation exists between the lateral compressive shear and the area of the compressive strut which formed between the two overlapping cracks. The axial capacity of the DEN specimens is greatly reduced by the presence of a lateral compressive shear. Self similar overlapping cracks were observed for all three specimen sizes. Thus it seems that the specimen size has no influence on the mode of failure, at least for the specimen ranges covered and pertinent to the load-path pursued. For the experiment with the proportional load-path (8/8s = 1.0), where tension and shear were applied simultaneously, a combination of failure modes exists. When the specimen size was changed from 200 m m to 50 mm, the failure mode changed from the formation of overlapping straight cracks to the growth of a network of inclined distributed cracks. The recently developed lattice model is capable of capturing the curved overlapping cracks found in these load-paths. The simulated peak load corresponds to the experimentally determined value in most cases, using material parameters in the model that were tuned to a uniaxial tensile test. Differences are observed in the postpeak regime. Yet those differences can be explained from the fact that small particles are excluded from the analyses [12] and from the fact that the analyses are carried out in two dimensions whereas the fracture process is in reality a threedimensional phenomenon. It seems that only a mode I criterion is sufficient for predicting crack growth under combined tension and shear.
Appendix. Concrete Composition
and Strength River gravel and sand: 0.125 to 0.25 mm, 303 kg/m3; 0.25 to 0.5 mm, 197 kg/m3; 0.5 to 1.0 mm, 515 kg/m3;
Biaxial Tension and Shear on Concrete
37
1.0 tO 2.0 ram, 500 kg/m3; Portland cement type B, 500 kg/m3; w/c = 0.5; for batch 46, fc = 49.66(2.23) MPa and fspl = 3.76(0.29) MPa; for batch 47, fc = 46.19(0.32) MPa and fspl = 3.78(0.23) MPa; and for batch 48, fc = 46.24(0.37) MPa and f~pl = 3.67(0.29) MPa.
References 1. Wittmann, F.H. Fracture Mechanics of Concrete; Wittmann, F.H., Ed.; Elsevier: Amsterdam, 1983; pp 43-74. 2. Hsu, T.T.C.; Slate, F.O.; Sturman, G.M.; Winter, G. J. Am. Conc. Inst. 1963, 2, 209-224. 3. Van Mier, J.G.M.; Nooru-Mohamed, M.B. Eng. Frac. Mech. 1990, 35, 617-628. 4. Li, B.; Maekawa, K.; Okamura, H. J. Fac. Eng. 1989, 40, 9-52. 5. Saouma, V.; Ayari, M.L.; Boggs, H.L. Fracture of Concrete and Rock; Shah, S.P.; Swartz, S.E., Eds.; Springer Verlag: New York, 1989; pp 311-333. 6. Hassanzadeh, M. Behaviour of Fracture Process Zones in Concrete Influenced by Simultaneously Applied Normal and Shear Displacements; Ph.D. Thesis, Lund Institute of Technology, Lund, Sweden, 1992. 7. Keuser, W.; Walraven, J. Fracture of Concrete and RockRecent Developments; Shah, S.P.; Swartz, S.E.; Barr, B.I.G., Eds.; Elsevier Applied Science Publishers: London, 1988; pp 625--634. 8. Reinhardt, H.W.; Cornelissen, H.A.W.; Hordijk, D.A. Fracture of Concrete and Rock; Shah, S.P.; Swartz, S.E., Eds.; Springer Verlag: New York, 1989; pp 117-130. 9. Van Mier, J.G.M.; Nooru-Mohamed, M.B.; Timmers, G. HERON 1991, 36, 1-104. 10. Nooru-Mohamed, M.B. Mixed-Mode Fracture of Concrete: An Experimental Approach; Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1992. 11. Schlangen, E.; Van Mier, J.G.M. Int. ]. Damage Mech. 1992, 1,435-454. 12. Schlangen, E. Experimental and Numerical Analysis of Fracture Processes in Concrete; Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1993. 13. De Borst, R. Non-Linear Analysis of Frictional Materials; Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1986. 14. Rots, J.G. Computational Modelling of Concrete Fracture; Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 1988.